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@ -119,3 +119,19 @@
year={2020},
publisher={Elsevier}
}
@book{violeau2012,
title={Fluid mechanics and the SPH method: theory and applications},
author={Violeau, Damien},
year={2012},
publisher={Oxford University Press}
}
@article{violeau2007,
title={Numerical modelling of complex turbulent free-surface flows with the SPH method: an overview},
author={Violeau, Damien and Issa, Reza},
journal={International Journal for Numerical Methods in Fluids},
volume={53},
number={2},
pages={277--304},
year={2007},
publisher={Wiley Online Library}
}

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@ -56,47 +56,59 @@ initiation. A parametrisation of waves depending on their source is also used to
on the type of scenario --- wave or tsunami. Those equations were later revised by \textcite{nandasena2011}, as they
were found to be partially incorrect. A revised formulation based on the same considerations was provided.
The assumptions on which \citeauthor{nott2003, nandasena2011} are based were then critisized by \textcite{weiss2015}. In
The assumptions on which \textcite{nott2003, nandasena2011} are based were then critisized by \textcite{weiss2015}. In
fact, according to them, the initiation of movement is not sufficient to guarantee block displacement.
\textcite{weiss2015} highlights the importance of the time dependency on block displacement. A method is proposed that
allows to find the wave amplitude that lead to block displacement.
allows to find the wave amplitude that lead to block displacement. Additionally, more recent research by
\textcite{lodhi2020} has shown that the equations proposed by \textcite{nott2003, nandasena2011} tend to overestimate
the minimum flow velocity needed to displace a block.
% Lack of observations -> observation
Whether it is \textcite{nott2003}, \textcite{nandasena2011} or \textcite{weiss2015}, all the proposed analytical
equations suffer from a major flaw; they are all based on simplified analytical models and statistical analysis.
Unfortunately, no block displacement event seems to have been observed directly in the past.
equations suffer from a major flaw: they are all based on very simplified analytical models and statistical analysis.
Unfortunately, no block displacement event seems to have been observed directly in the past, and those events are
difficult to predict.
In this paper, we study such an event. On February 28, 2017, a \SI{50}{\tonne} concrete block was dropped by a wave on
the crest of the Artha breakwater. Luckily, the event was captured by a photographer, and a wave buoy located
\SI{1.2}{\km} offshore captured the seastate. Information from the photographer allowed to establish the approximate
time at which the block displacement occured. The goal of this paper is to model the hydrodynamic conditions near the
breakwater that lead to the displacement of the \SI{50}{\tonne} concrete block.
the crest of the Artha breakwater (Figure~\ref{fig:photo}). Luckily, the event was captured by a photographer, and a
wave buoy located \SI{1.2}{\km} offshore captured the seastate. Information from the photographer allowed to establish
the approximate time at which the block displacement occured. The goal of this paper is to model the hydrodynamic
conditions near the breakwater that lead to the displacement of the \SI{50}{\tonne} concrete block.
% Modeling flow accounting for porous media
Several approaches can be used when modelling flow near a breakwater. Depth-averaged models can be used to study the
transformation of waves on complex bottoms. Studying the hydrodynamic conditions under the surface can be achieved using
smoothed-particles hydrodynamics (SPH) or volume of fluid (VOF) models. SPH models rely on a Lagrangian representation
of the fluid, while VOF models rely on an Eulerian representation. VOF models are generally more mature for the study of
multiphase incompressible flows.
transformation of waves on complex bottoms. Studying the hydrodynamic conditions under the surface can be achieved
using smoothed-particles hydrodynamics (SPH) or volume of fluid (VOF) models. SPH models rely on a Lagrangian
representation of the fluid \parencite{violeau2012}, while VOF models rely on an Eulerian representation. VOF models
are generally more mature for the study of multiphase incompressible flows, while SPH models generally require more
processing power for similar results \parencite{violeau2007}.
In this paper, we first use a one-dimensionnal depth-averaged non-linear non-hydrostatic model to verify that the signal
measured by the wave buoy can be used as an incident wave input for the determination of hydrodynamic conditions near
the breakwater. For this model, we use a SWASH model \parencite{zijlema2011} already calibrated by \textcite{poncet2021}
on a domain reaching \SI{1450}{\m} offshore of the breakwater.
In this paper, we first use a one-dimensionnal depth-averaged non-linear non-hydrostatic model to verify that the
signal measured by the wave buoy can be used as an incident wave input for the determination of hydrodynamic conditions
near the breakwater. For this model, we use a SWASH model \parencite{zijlema2011} already calibrated by
\textcite{poncet2021} on a domain reaching \SI{1450}{\m} offshore of the breakwater.
Then, we use a nested VOF model in two vertical dimensions that uses the output from the larger scale SWASH model as
initial and boundary conditions to obtain the hydrodynamic conditions on the breakwater. The models uses olaFlow
initial and boundary conditions to obtain the hydrodynamic conditions on the breakwater. The model uses olaFlow
\parencite{higuera2015}, a VOF model based on volume averaged Reynolds averaged Navier-Stokes (VARANS) equations, and
which relies on a macroscopic representation of the porous armour of the breakwater. The model is qualitatively
calibrated using photographs from the storm of February 28, 2017. Results from the nested models are finally compared to
the analytical equations provided by \textcite{nandasena2011}.
calibrated using photographs from the storm of February 28, 2017. Results from the nested models are finally compared
to the analytical equations provided by \textcite{nandasena2011}.
\begin{figure*}
\centering
\includegraphics[height=.4\textwidth]{fig/pic1.jpg}
\includegraphics[height=.4\textwidth]{fig/pic2.jpg}
\caption{Photographs taken during and after the wave that displaced a \SI{50}{\tonne} concrete block onto the Artha
breakwater.}\label{fig:photo}
\end{figure*}
\section{Results}
\subsection{Identified wave}
Preliminary work with the photographer allowed to identify the time at which the block displacement event happened.
Using the data from the wave buoy located \SI{1250}{\m} offshore of the Artha breakwater, a seamingly abnormally large
wave of \SI{14}{\m} amplitude was identified that is supposed to have lead to the block displacement.
wave of \SI{14}{\m} amplitude was identified that is supposed to have led to the block displacement.
Initial analysis of the buoy data plotted in Figure~\ref{fig:wave} shows that the movement of the buoy follows two
orbitals that correspond to an incident wave direction. These results would indicate that the identified wave is
@ -115,14 +127,10 @@ with a period of over \SI{30}{\s}.
\subsection{Reflection analysis}
The results from the large scale SWASH model using two configurations --- one of them being the real bathymetry, and the
other being a simplified bathymetry without the breakwater --- are compared in Figure~\ref{fig:swash}. The results
obtained with both simulations show a maximum wave amplitude of \SI{13.9}{\m} for the real bathymetry, and \SI{12.1}{\m}
in the case where the breakwater is removed.
The 13\% difference between those values highlights the existence of a notable amount of reflection at the buoy.
Nonetheless, the gap between the values is still fairly small and the extreme wave identified on February 28, 2017 at
17:23:08 could still be considered as an incident wave.
The results from the large scale SWASH model using two configurations --- one of them being the real bathymetry, and
the other being a simplified bathymetry without the breakwater --- are compared in Figure~\ref{fig:swash}. The results
obtained with both simulations show a maximum wave amplitude of \SI{13.9}{\m} for the real bathymetry, and
\SI{12.1}{\m} in the case where the breakwater is removed.
\begin{figure*}
\centering
@ -146,12 +154,10 @@ crest increases, with a zone reaching \SI{400}{\m} long in front of the wave whe
qualitatively estimated position of the wave front.}\label{fig:swash_trans}
\end{figure*}
\subsection{Wavelet analysis}
In an attempt to understand the identified wave, a wavelet analysis is conducted on raw buoy data as well as at
different points along the SWASH model using the method proposed by \textcite{torrence1998}. The results are displayed
in Figure~\ref{fig:wavelet} and Figure~\ref{fig:wavelet_sw}. The wavelet power spectrum shows that the major component
in the identified wave is a high energy infragravity wave, with a period of around \SI{60}{\s}.
in identified rogue waves is a high energy infragravity wave, with a period of around \SI{60}{\s}.
The SWASH model seems to indicate that the observed transformation of the wave can be characterized by a transfer of
energy from the infragravity band to shorter waves from around \SI{600}{\m} to \SI{300}{\m}, and returning to the
@ -159,8 +165,9 @@ infragravity band at \SI{200}{\m}.
\begin{figure*}
\centering
\includegraphics{fig/wavelet9312.pdf}
\caption{Normalized wavelet power spectrum from the raw buoy timeseries.}\label{fig:wavelet}
\includegraphics{fig/wavelet.pdf}
\caption{Normalized wavelet power spectrum from the raw buoy timeseries for identified rogue waves on february 28,
2017.}\label{fig:wavelet}
\end{figure*}
\begin{figure*}
\centering
@ -172,16 +179,17 @@ infragravity band at \SI{200}{\m}.
The two-dimensionnal olaFlow model near the breakwater allowed to compute the flow velocity near and on the breakwater
during the passage of the identified wave. The results displayed in Figure~\ref{fig:U} show that the flow velocity
reaches a maximum of \SI{14.5}{\m\per\s} towards the breakwater during the identified extreme wave. Although the maximum
reached velocity is slightly lower than earlier shorter waves (at $t=\SI{100}{\s}$ and $t=\SI{120}{\s}$, with a maximum
velocity of \SI{17.3}{\m\per\s}), the flow velocity remains high for twice as long as during those earlier waves. The
tail of the identified wave also exhibits a water level over \SI{5}{\m} for over \SI{40}{\s}.
reaches a maximum of \SI{14.5}{\m\per\s} towards the breakwater during the identified extreme wave. Although the
maximum reached velocity is similar to earlier shorter waves, the flow velocity remains high for twice as long as
during those earlier waves. The tail of the identified wave also exhibits a water level over \SI{5}{\m} for over
\SI{40}{\s}.
\begin{figure*}
\centering
\includegraphics{fig/U.pdf}
\caption{Horizontal flow velocity computed with the olaFlow model at $x=\SI{-20}{\m}$ on the breakwater armor. The
identified wave reaches this point around $t=\SI{175}{\s}$.}\label{fig:U}
\caption{Horizontal flow velocity computed with the olaFlow model at $x=\SI{-20}{\m}$ on the breakwater armor.
Bottom: horizontal flow velocity at $z=\SI{5}{\m}$. The identified wave reaches this point around
$t=\SI{175}{\s}$.}\label{fig:U}
\end{figure*}
\section{Discussion}
@ -194,12 +202,19 @@ twice the significant wave height over a given period. The identified wave fits
rogue waves often occur from non-linear superposition of smaller waves. This seems to be what we observe on
Figure~\ref{fig:wave}.
The wavelet power spectrum shows that a very prominent infragravity component is present, which usually corresponds to
non-linear interactions of smaller waves. \textcite{dysthe2008} mentions that such waves in coastal waters are often the
result of refractive focusing. On February 28, 2017, the frequency of rogue waves was found to be of 1 wave per 1627,
which is considerably more than the excedance probability of 1 over 10\textsuperscript4 calculated by
\textcite{dysthe2008}. Additionnal studies should be conducted to understand focusing and the formation of rogue waves
in front of the Saint-Jean-de-Luz bay.
As displayed in Figure~\ref{fig:wavelet}, a total of 4 rogue waves were identified on february 28, 2017 in the raw buoy
timeseries using the wave height criteria proposed by \textcite{dysthe2008}. The wavelet power spectrum shows that a
very prominent infragravity component is present, which usually corresponds to non-linear interactions of smaller
waves. \textcite{dysthe2008} mentions that such waves in coastal waters are often the result of refractive focusing. On
February 28, 2017, the frequency of rogue waves was found to be of 1 wave per 1627, which is considerably more than the
excedance probability of 1 over 10\textsuperscript4 calculated by \textcite{dysthe2008}. Additionnal studies should be
conducted to understand focusing and the formation of rogue waves in front of the Saint-Jean-de-Luz bay.
An important point to note is that rogue waves are often short-lived: their nature means that they often separate into
shorter waves shortly after appearing. A reason for which such rogue waves can be maintained over longer distances can
be a change from a dispersive environment such as deep water to a non-dispersive environment. The bathymetry near the
wave buoy (Figure~\ref{fig:bathy}) shows that this might be what we observe here, as the buoy is located near a step in
the bathymetry, from around \SI{40}{\m} to \SI{20}{\m} depth.
\subsection{Reflection analysis}
@ -207,10 +222,11 @@ The 13\% difference between those values highlights the existence of a notable a
Nonetheless, the gap between the values is still fairly small and the extreme wave identified on February 28, 2017 at
17:23:08 could still be considered as an incident wave.
Unfortunately, the spectrum wave generation method used by SWASH could not reproduce simlar waves to the one observed at
the buoy. As mentionned by \textcite{dysthe2008}, such rogue waves cannot be deterministicly from the wave spectrum. For
this reason, this study only allows us to observe the influence of reflection on short waves, while mostly ignoring
infragravity waves.
Unfortunately, the spectrum wave generation method used by SWASH could not reproduce simlar waves to the one observed
at the buoy. As mentionned by \textcite{dysthe2008}, such rogue waves cannot be deterministicly from the wave spectrum.
For this reason, this study only allows us to observe the influence of reflection on short waves, while mostly ignoring
infragravity waves. Those results are only useful if we consider that infragravity waves behave similarly to shorter
waves regarding reflection.
\subsection{Wave transformation}
@ -236,11 +252,16 @@ Those results tend to confirm recent research by \textcite{lodhi2020}, where it
threshold tend to overestimate the minimal flow velocity needed for block movement, although further validation of the
model that is used would be needed to confirm those findings.
Additionally, the flow velocity that is reached during the identified wave is not the highest that is reached in the
model. Other shorter waves yield similar flow velocities on the breakwater, but in a smaller timeframe. The importance
of time dependency in studying block displacement would be in accordance with research from \textcite{weiss2015}, who
suggested that the use of time-dependent equations for block displacement would lead to a better understanding of the
phenomenon.
Additionally, similar flow velocities are reached in the model. Other shorter waves yield similar flow velocities on
the breakwater, but in a smaller timeframe. The importance of time dependency in studying block displacement would be
in accordance with research from \textcite{weiss2015}, who suggested that the use of time-dependent equations for block
displacement would lead to a better understanding of the phenomenon.
Although those results are a major step in a better understanding of block displacement in coastal regions, further
work is needed to understand in more depth the formation and propagation of infragravity waves in the near-shore
region. Furthermore, this study was limited to a single block displacement event, and further work should be done to
obtain more measurements and observations of such events, although their rarity and unpredictability makes this task
difficult.
\section{Methods}
@ -261,14 +282,23 @@ over \SI{0.2}{\Hz}.
All wavelet analysis in this study is conducted using a continuous wavelet transform over a Morlet window. The wavelet
power spectrum is normalized by the variance of the timeseries, following the method proposed by
\textcite{torrence1998}.
\textcite{torrence1998}. This analysis extracts a time-dependent power spectrum and allows to identify the composition
of waves in a time-series.
\subsection{SWASH models}
\subsubsection{Domain}
\begin{figure}
\centering
\includegraphics{fig/bathy2d.pdf}
\caption{Bathymetry in front of the Artha breakwater. The extremities of the line are the buoy and the
breakwater.}\label{fig:bathy}
\end{figure}
A \SI{1750}{\m} long domain is constructed in order to study wave reflection and wave transformation over the bottom
from the wave buoy to the breakwater. Bathymetry with a resolution of around \SI{1}{\m} was used for most of the domain.
from the wave buoy to the breakwater. Bathymetry with a resolution of around \SI{1}{\m} was used for most of the domain
(Figure~\ref{fig:bathy}).
The breakwater model used in the study is taken from \textcite{poncet2021}. A smoothed section is created and considered
as a porous media in the model.
@ -322,12 +352,12 @@ the SWASH model, the porous armour was considered at a macroscopic scale.
A volume-of-fluid (VOF) model in two-vertical dimensions based on volume-averaged Reynolds-averaged Navier-Stokes
(VARANS) equations is used (olaFlow, \cite{higuera2015}). The model was initially setup using generic values for porous
breakwater studies. A sensibility study conducted on the porosity parameters found a minor influence of these values on
the final results.
breakwater studies. A sensibility study conducted on the porosity parameters found the influence of these values on
the final results to be very minor.
The k-ω SST turbulence model was used, as it produced much more realistic results than the default k-ε model, especially
compared to the photographs from the storm of February 28, 2017. The k-ε model yielded very high viscosity and thus
strong dissipation in the entire domain, preventing an accurate wave breaking representation.
The k-ω SST turbulence model was used, as it produced much more realistic results than the default k-ε model,
especially compared to the photographs from the storm of February 28, 2017. The k-ε model yielded very high viscosity
and thus strong dissipation in the entire domain, preventing an accurate wave breaking representation.
\subsubsection{Boundary conditions}